The present invention relates generally to the field of using ultrasonic waves to non-destructively detect structural characteristics of work pieces. More particularly, this invention relates to a method and apparatus for focusing ultrasonic waves that are propagated from a phased array to a spherically-bounded object so that the waves arrive at a pre-selected focal point at the same time and in phase and thereby significantly improve, nondestructively, the detectability of possible flaws in and structural characteristics of the object. Still more particularly, this invention relates to a method and apparatus for non-destructively evaluating spherically bounded objects, such as reactor pressure vessels, ball bearings, and the like by obtaining by iteration of a focal law equation according to the invention which follows a path on the surface of the spherically bounded object.